For some reason internet comics following a continuous story are a little lesser known. I suppose it's because when you send someone a comic link you usually want it to be short and funny. That doesn't mean the long term comics can't be funny, i just means there's a bit of impatience when it comes to internet comics.
Here are a few series that either continue the story with a punch line on every page or are just so awesome it's worth being patient:
Power Puff Girls Doujinshi:
During the last comicon Cartoon Network was giving out mouse pads to a new online game series. I walked up to the sales lady and said: "Power Puff Doujinshi!"
Her reply was: "No it's not Power Puff Doujinshi, but your the nineth person to say that to me today. Who is this bleedman guy?"
Long before someone was hired by cartoon network to work on the online game, similar designs were created for Bleedman's online comic Power Puff Doujinshi. Explore the page and you'll find even more parodies.

You'd expect Scott to have his hands full with VGcats but his newer series, Super Effective, packs just as much parody while following the Pokemon Kanto storyline. There's a joke on every page but the connection to the Pokemon Series has you wondering what kind of take he'll have on the next event. While most of the jokes expect you to have played Pokemon they're are several you'll get while being completely oblivious.

Dr. McNinja:
"Is the doctor fighting Mexicans on raptors? He is..! He is fighting Mexicans on Raptors. There are patients waiting to be seen. I ought to give him a piece of my mind!"
Dr. Mcninja started on a mission to stop Ronald McDonald's secret plan for world domination. A strange series that flops back and forth between somewhat making sense and being randomly awesome, Dr. McNinja is definitely up there as an Internet Literary Classic.

Didoz,
The Neon Wind